Simplifying Combination Formula: \binom{n}{\frac{n}{2}} using Factorial Formula

In summary, the conversation discusses the simplification of the combination \binom{n}{\frac{n}{2}} and the use of Stirling's formula to characterize its behavior as n becomes larger. The original formula is expressed as 2^{n}(1-\frac{1}{n})(1-\frac{1}{n-2})(1-\frac{1}{n-4})... and the goal is to simplify it. The conversation also mentions the importance of familiarity with Stirling's formula in probability.
  • #1
Aleoa
128
5
Member has been warned not to delete the template.
I'm trying to simplify the combination defined as : [itex]\binom{n}{\frac{n}{2}}[/itex].

I did some calculations, starting from the factorial formula [itex]\frac{n!}{(\frac{n}{2})!(\frac{n}{2})!}[/itex] and i found this form :

[itex]2^{n}(1-\frac{1}{n})(1-\frac{1}{n-2})(1-\frac{1}{n-4})...[/itex]

but i don't know how to continue, can you help me ?
 
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  • #2
Aleoa said:
but i don't know how to continue, can you help me ?
No, since you haven't said what your goal is. To me ##\binom{n}{\frac{n}{2}}## is already fine.
 
  • #3
I want to characterize the behaviour of the formula as n become larger, so I'm trying to simplify it . I'm sorry for the template, next time i'll write it correctly. Thanks for the support
 
  • #4
Aleoa said:
I want to characterize the behaviour of the formula as n become larger, so I'm trying to simplify it . I'm sorry for the template, next time i'll write it correctly. Thanks for the support
In this case I'd try where Stirling's approximation would get me.
 
  • #5
Aleoa said:
I want to characterize the behaviour of the formula as n become larger, so I'm trying to simplify it . I'm sorry for the template, next time i'll write it correctly. Thanks for the support

As fresh_42 suggested, use Stirling's formula. Every student of probability should be thoroughly familiar with that formula, as it is used everywhere.
 

What is a combination?

A combination is a selection of items from a larger set without regard to the order in which they are chosen. For example, if you have 5 different colored balls and you choose 3 of them, the different combinations could be red, blue, green or blue, green, yellow, etc.

How do you simplify a combination?

To simplify a combination, you need to use the formula nCr = n! / r!(n-r)!, where n represents the total number of items in the set and r represents the number of items you are choosing. This will give you the total number of combinations possible.

What is the difference between a combination and a permutation?

A combination is a selection of items without regard to order, while a permutation is a selection of items with regard to order. For example, choosing 3 items from a set of 5 without regard to order would be a combination, while choosing 3 items in a specific order would be a permutation.

What is the importance of simplifying combinations?

Simplifying combinations allows you to easily calculate the total number of possible combinations, which is useful in many fields such as mathematics, computer science, and statistics. It also helps to avoid repetition and ensure accuracy in calculations.

What are some real-life applications of combinations?

Combinations are used in various fields such as genetics, probability, and cryptography. In genetics, combinations are used to determine the probability of inheriting certain genes. In probability, combinations are used to calculate the likelihood of certain events occurring. In cryptography, combinations are used to create secure passcodes.

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